- Euler's Formula relates sinusoidal functions and a complex exponential by:

- From this expression, we see that a complex exponential can provide an alternative representation for a real sinusoidal signal:

- Complex exponentials are convenient for calculating multiplications or divisions of sinusoidal signals:

- A complex signal
can be rewritten
.
- We then define
as the complex amplitude, polar representation of amplitude and phase shift of the complex exponential signal. This term also represents a vector in the complex plane where
*A*= |X| and . - We then have
. The
term has magnitude = 1 and can be interpreted as adding an incremental angle or phase shift to
*X*every time step*t*. - In this way,
*X*is a phasor rotating in the complex plane with radian frequency .

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